Properties

Degree 2
Conductor 41
Sign $0.884 + 0.466i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.00 − 1.38i)2-s + (2.22 + 2.22i)3-s + (−0.290 + 0.892i)4-s + (−2.57 − 0.837i)5-s + (0.845 − 5.33i)6-s + (−0.252 − 1.59i)7-s + (−1.73 + 0.562i)8-s + 6.93i·9-s + (1.43 + 4.41i)10-s + (0.314 − 0.617i)11-s + (−2.63 + 1.34i)12-s + (−1.50 − 0.238i)13-s + (−1.95 + 1.95i)14-s + (−3.87 − 7.60i)15-s + (4.04 + 2.93i)16-s + (0.942 + 0.480i)17-s + ⋯
L(s)  = 1  + (−0.712 − 0.980i)2-s + (1.28 + 1.28i)3-s + (−0.145 + 0.446i)4-s + (−1.15 − 0.374i)5-s + (0.345 − 2.17i)6-s + (−0.0953 − 0.602i)7-s + (−0.611 + 0.198i)8-s + 2.31i·9-s + (0.453 + 1.39i)10-s + (0.0948 − 0.186i)11-s + (−0.760 + 0.387i)12-s + (−0.418 − 0.0662i)13-s + (−0.522 + 0.522i)14-s + (−1.00 − 1.96i)15-s + (1.01 + 0.734i)16-s + (0.228 + 0.116i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $0.884 + 0.466i$
motivic weight  =  \(1\)
character  :  $\chi_{41} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 41,\ (\ :1/2),\ 0.884 + 0.466i)$
$L(1)$  $\approx$  $0.664232 - 0.164493i$
$L(\frac12)$  $\approx$  $0.664232 - 0.164493i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 41$,\(F_p(T)\) is a polynomial of degree 2. If $p = 41$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad41 \( 1 + (-5.96 + 2.31i)T \)
good2 \( 1 + (1.00 + 1.38i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (-2.22 - 2.22i)T + 3iT^{2} \)
5 \( 1 + (2.57 + 0.837i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.252 + 1.59i)T + (-6.65 + 2.16i)T^{2} \)
11 \( 1 + (-0.314 + 0.617i)T + (-6.46 - 8.89i)T^{2} \)
13 \( 1 + (1.50 + 0.238i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (-0.942 - 0.480i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (0.126 - 0.0201i)T + (18.0 - 5.87i)T^{2} \)
23 \( 1 + (-1.52 + 1.10i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-7.39 + 3.76i)T + (17.0 - 23.4i)T^{2} \)
31 \( 1 + (-0.373 - 1.14i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.86 - 5.75i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (6.21 + 8.55i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (0.991 - 6.26i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (11.1 - 5.67i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (6.61 - 4.80i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.54 - 2.13i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.85 + 7.56i)T + (-39.3 + 54.2i)T^{2} \)
71 \( 1 + (-1.94 + 3.82i)T + (-41.7 - 57.4i)T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 + (8.14 + 8.14i)T + 79iT^{2} \)
83 \( 1 - 6.49T + 83T^{2} \)
89 \( 1 + (-0.451 - 2.84i)T + (-84.6 + 27.5i)T^{2} \)
97 \( 1 + (-0.548 - 1.07i)T + (-57.0 + 78.4i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.86034034047056545998634846908, −15.09192268169557184243041101400, −13.91668135831896817286689648704, −12.15279797673577385731276539205, −10.82266136255820485400926932944, −9.966586388776865291630093942822, −8.840787118948823419225175922524, −7.929981449953059313345546284518, −4.41062707061487130824974387030, −3.11011041795528540377520344850, 3.10376594578513509992757684009, 6.60477792903251132341173906885, 7.54047262379860793891415335186, 8.279738127154091636728314544949, 9.330940758588791588273949578440, 11.86767606752282614219142286858, 12.70559529206134745781980388128, 14.35979216609744429741411765849, 15.07918788913189147572613976671, 16.00196789900821185331994805615

Graph of the $Z$-function along the critical line